Toward the Determination of Effective Action in Superstring Theory and M-

Theory

Yoshifumi Hyakutake (Osaka Univ.)

1. 1. IntroductionIntroduction• Divergences in 4 dim. quantum

gravityIt is important to formulate gravitational interaction in a way consistent with quantum mechanics.

Perturbative approach to quantum gravity, however, is problematic. Loop corrections give divergences in UV.

Non-susy gravity: divergence at 1 or 2 loop Supergravity : divergence at 3 loop(except

N=8)

• Superstring theory as a quantum gravityIn order to cure the divergences, we consider not

a point particle object but a string object.

Then interactions are smeared around the string scale, and no UV divergence appears in superstring theory.

t’Hooft, Veltman

• Superstring theory in low energy region

Superstring theory is perturbatively defined around 10 dim. flat space-time.

And the low energy limit of superstring theory is described by supergravity.

Superstring

Supergravity

Loop calculations in superstring theory give corrections to the low energy supergravity.

• Quantum correction to supergravity

Superstring

Supergravity + higher derivative corrections

Yoneya

Gross, Witten

And higher derivative corrections in string theories are considerably investigated in various ways

• String scattering amplitude

• Non linear sigma model

• Superfield method

• Duality

• Noether’s method … and so on

Since corrections become important to analyze black hole physics or singularities in classical gravity.

I will review the current knowledge on the higher derivative terms in string theory and M-theory.

I also discuss the recent progress on the finiteness of N=8 supergravity.

1. Introduction

2. Effective Action from Superstring Amplitude

3. Effective Action from Local Supersymmetry

4. Perturbative and Non-perturbative Terms via Duality

5. Non-renormalization Conditions in Type IIA

6. SYM vs. SUGRA

7. Summary

Plan :

2. Effective Action from Superstring Amplitude

Closed oriented superstring amplitudes can be evaluated by inserting vertex operators of external states on Riemann surfaces.

sugra

0

sugra

0 0 0

stringy

quantum

Nontrivial corrections start from 4pt amplitude

• Corrections in Type II superstring

• Evaluation of 4pt Graviton Amplitude

Momentum and two (left and right) polarization vectors are assigned at each external leg.

Green, SchwarzGross, WittenGross, Sloan

The same kinematical factor arises for tree and 1-loop.Here ( ) is a kinematical factor for left (right) mover.

: SO(8) generator

4pt graviton amplitude is given by

where and are functions of Mandelstam variables

Constant terms of and become coefficients of

Remark : Others will give terms with more derivatives

By the analysis so far, bosonic part in type II is written as

M-theory is described by 11 dim. N=1 Supergravity.

We will check these forms by local supersymmetry

• Effective Action in type II

• Effective Action in M

Higher derivative corrections in M-theory can be obtained by lifting the IIA result.

Cremmer, Julia, Scherk

3. 3. Effective Action from Local Supersymmetry

Since perturbative methods in M-theory are not developed, it is impossible to obtain higher derivative terms by evaluating scattering amplitudes of membranes. The best way to determine this structure is to use the invariance under local supersymmetry

Noether method + computer programming

Here we mainly concentrate on bosonic terms and consider cancellation of O(1) and O(F) terms step by step.

O(1) O(F) O(F^2) …

O( )

O( )

Variation

• Strategy

Hyakutake, Ogushi

By using computer program, we obtain 7 independent terms.

Q. How many AR^4 terms ?

There are two terms

Q. How many R^4 terms ?

The cancellation mechanism up to O(F) is sketched as

In order to cancel variations of these bosonic terms, it is necessary to add fermionic terms to the ansatz

Variations of the ansatz are expanded by the following terms

244 Equations among 126 Variables

• Cancellation

After miraculous cancellation, the bosonic part is determined as

The first term corresponds to tree level and the second does to one-loop part in type IIA superstring

• Two superinvariants

tree

1-loop

• The next step is to examine the invariance under local supersymmetry up to O(F^2)

In order to execute the cancellation to this order, we have to add

The variations of this ansatz are expanded by

The cancellation mechanism up to O(F^2) is sketched as

4169 Equations among 1544 Variables

From this cancellation we obtain

The structure of R^4 terms is completely determined by local supersymmetry

1-loop

• Vanishing theorem

Tree and one-loop amplitudes only contribute to terms

Proof :

In 11 dim. there is only one superinvariant which contain terms. These become tree level or 1-loop terms in type IIA by Kaluza-Klein reduction. No terms more than one-loop.

11d 10d

Sum of KK non-zero modesKK zero mode

Green, Gutperle, Vanhove

4. Perturbative and Non-perturbative Terms via Duality

Let us reexamine 1-loop amplitude for 4 gravitons in the low energy limit

This expression contains the integral of loop momentum, and it is implicitly included in

In order to lift this to 11 dimensions, the sum of KK momentum should appear.

• 11 dim. 1-loop amplitude on a circle

Green, Gutperle, Vanhove

Russo, Tseytlin

Momentum along 9th dimension should be discretized.

• 11 dim. 1-loop amplitude on a torus

10 dimensional type IIB is realized if

is a non-holomorphic Eisenstein series

D-instanton

SummarSummaryy

Higher derivative corrections in Type II and M-theory are considered via

• String scattering amplitude

• Local Supersymmetry

• Duality

Recent Arguments on the finiteness of N=8 Supergravity

Yoshifumi Hyakutake (Osaka Univ.)

Calculation of 11 dim. L-loop amplitude is difficult. By using power counting, however, we can restrict its form. The L-loop amplitude on a torus which includes term will be

• 11 dim. L-loop amplitude on a torus

where

derivatives

subdivergences

Green, Russo, Vanhove

And following constraints are imposed

5. Non-renormalization Conditions in 5. Non-renormalization Conditions in Type IIAType IIA

By considering the duality between M and IIA,

Effective action in IIA from genus can be derived

• Higher derivative action in type IIA

Terms which are linear to survive after decompactification

or

gives

Then we obtain strong conditions.

• Conditions for higher derivative terms in type IIA

h-loop amplitude in type IIA contributes as follows.

1. No contributions to

2. can be determined by 1-loop in 11 dim.

3. are permitted and may arise from L-loop in 11 dim. (L>1)

Leading term in low energy can be given by

• Dimensional reduction

If the same property holds for lower dimensions, the leading contributions can be given by

If this is true, UV divergences are absent in dimensions which satisfy

4 dim. N=8 supergravity seems to be UV finite

6. SYM vs. SUGRA6. SYM vs. SUGRA

• Tree level (KLT relation)

Kawai, Lewellen, Tye

Tree level 4 graviton amplitude is expressed as a ‘‘product’’ of two tree level 4 point gluon amplitudes.

By using the relation of gamma functions,

we obtain KLT relation (closed)~(open)^2

Low energy

1-loop

• 1-loop 4pt in the low energy limit

Green, Schwarz, Brink

Tree level 4pt and 1-loop 4pt amplitudes share the same kinematical factor. Then the Low energy limit of 4pt 1-loop amplitudes become

Here is given by the scalar box integral

UV divergence arises in 8 dimensions

SUGRA~(SYM)^2

• 2-loop 4pt in N=4 SYMBern, Rozowsky, Yan

Bern, Dixon, Dunbar, Perelstein, Rozowsky2-loop 4pt amplitudes for N=4 SYM can be computed in

terms of scalar integral functions via cutting method.

Calculations are done by employing spinor helicity formalism

UV divergence arises in 7 dimensions

Rung rule

• 2-loop 4pt in N=8 SUGRABern, Rozowsky, Yan

Bern, Dixon, Dunbar, Perelstein, Rozowsky2-loop 4pt amplitudes for N=8 SUGRA can be obtained

by squaring the SYM factor

UV divergence arises in 7 dimensions

Rung rule

• Estimation of UV divergences by rung rule

Let us evaluate L-loop diagram

• N=4 SYM

• N=8 SUGRA

Divergence free

Divergence free?

• 3-loop calculation tells …

4pt graviton amplitude at 3-loop is divergent in 6 dimension. This is the same as N=4 SYM. Thus the conjecture is

for N=8 SUGRA.

It seems that D=4 N=8 SUGRA is UV finite.

7. Summary7. Summary

Higher derivative corrections in Type II and M-theory are considered via

• String scattering amplitude

• Local Supersymmetry

• Duality

Finiteness of N=8 supergravity is reviewed